Gaussian processes (Kriging) are interpolating data-driven models that are frequently applied in various disciplines. Often, Gaussian processes are trained on datasets and are subsequently embedded as surrogate models in optimization problems. These optimization problems are nonconvex and global optimization is desired. However, previous literature observed computational burdens limiting deterministic global optimization to Gaussian processes trained on few data points. We propose a reduced-space formulation for deterministic global optimization with trained Gaussian processes embedded. For optimization, the branch-and-bound solver branches only on the free variables and McCormick relaxations are propagated through explicit Gaussian process models. The approach also leads to significantly smaller and computationally cheaper subproblems for lower and upper bounding. To further accelerate convergence, we derive envelopes of common covariance functions for GPs and tight relaxations of acquisition functions used in Bayesian optimization including expected improvement, probability of improvement, and lower confidence bound. In total, we reduce computational time by orders of magnitude compared to state-of-the-art methods, thus overcoming previous computational burdens. We demonstrate the performance and scaling of the proposed method and apply it to Bayesian optimization with global optimization of the acquisition function and chance-constrained programming. The Gaussian process models, acquisition functions, and training scripts are available open-source within the “MeLOn—MachineLearning Models for Optimization” toolbox (https://git.rwth-aachen.de/avt.svt/public/MeLOn).

We recently demonstrated the potential of deterministic global optimization in a reduced-space formulation for flowsheet optimization. However, the consideration of implicit unit operations such as flash calculations is still challenging and the solution of complex flowsheets incorporating such operations can be intractable. We show that the solution of flash equations can be integrated in global optimization via artificial neural networks (ANNs). Thus, flash calculations are no longer performed within the flowsheet optimization. Instead, flash equations are solved offline and then learned using ANNs. ANNs have been used successfully in the literature to learn flash equilibria but have not yet been included in deterministic global optimization for this task. We embed the ANNs in a hybrid model and use deterministic global optimization to solve it. In addition, we utilize deterministic global optimization to calculate a guaranteed worst-case accuracy of ANNs compared to a rigorous model. We demonstrate the proposed approach on an illustrative five-component vapor-liquid equilibrium flash using our in-house solver MAiNGO.

beta version is available open-source works well for problems formulated in reduced space runs on parallel computing can be extended to the user's needs can be used as general-purpose solver.